\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
E \arrow[r, "u"] \arrow[d] & F \arrow[d] \arrow[dl, "v'"'] \\
E_0 \arrow[r, bend left=18, "u_0"] & F_0 \arrow[l, bend left=18, "v_0"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
E \arrow[r, "u"] \arrow[d] \arrow[dr] & F \arrow[d] \arrow[dl] \\
E_0 & F_0
\end{tikzcd}

\begin{tikzcd}[row sep=small]
E \arrow[d, bend right=25, "\alpha'"'] \\
E_1 \arrow[u, bend right=25, "\alpha"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
E \arrow[rr, "u"] \arrow[dr, "\alpha'"'] & & F \\
& E_1 \arrow[ur, "\beta"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\Phi_2(C) \arrow[r, "\pi\vert\Phi_2"] & C \\
C \times C \arrow[u] \arrow[ur, "\otimes"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
L \otimes L' \arrow[r, "\varphi_{L,L'}"] \arrow[d, "s_{L,L'}"'] & \mathbf{1} \\
L' \otimes L \arrow[ur, "\varphi_{L',L}"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
L \otimes M \arrow[r, "s_{L,M}"] \arrow[d, "L \otimes q"'] &
M \otimes L \arrow[d, "q \otimes L"] \\
L \otimes \mathbf{1} \arrow[r, "s_{L,\mathbf{1}}"'] & \mathbf{1} \otimes L
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\mathcal{T} \arrow[r] \arrow[d] & \mathcal{T}^{(I)} \arrow[d] \\
\mathcal{T}_0 \arrow[r] & \mathcal{T}_0^{(I)}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\overline{C} \arrow[dr, "\overline{\varphi}"] & \\
C \arrow[u] \arrow[r, "\varphi"'] & \mathrm{Ens}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\psi \arrow[r] \arrow[dr] & \psi' \arrow[d, "\exists"] \\
& \varphi
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\varphi \arrow[r] \arrow[dr] & \varphi \sqcup \psi \arrow[d, "\exists"] \\
& \varphi
\end{tikzcd}